Surveying 18 12CO-bright galaxies from the ATLAS 3D early-type galaxy sample with the Institut de Radio Astronomie Millimetrique (IRAM) 30-m telescope, we detect 13CO(1-0) and 13CO(2-1) in all 18 galaxies, HCN(1-0) in 12/18 and HCO +(1-0) in 10/18. We find that the line ratios 12CO(1-0)/ 13CO(1-0) and 12CO(1-0)/HCN(1-0) are clearly correlated with several galaxy properties: total stellar mass, luminosity-weighted mean stellar age, molecular-to-atomic gas ratio, dust temperature and dust morphology. We suggest that these correlations are primarily governed by the optical depth in the 12CO lines; interacting, accreting and/or starbursting early-type galaxies have more optically thin molecular gas while those with settled dust and gas discs host optically thick molecular gas. The ranges of the integrated line intensity ratios generally overlap with those of spirals, although we note some outliers in the 12CO(1-0)/ 13CO(1-0), 12CO(2-1)/ 13CO(2-1) and HCN/HCO +(1-0) ratios. In particular, three galaxies are found to have very low 12CO(1-0)/ 13CO(1-0) and 12CO(2-1)/ 13CO(2-1) ratios. Such low ratios may signal particularly stable molecular gas which creates stars less efficiently than 'normal' (i.e. below Schmidt-Kennicutt prediction), consistent with the low dust temperatures seen in these galaxies.